A conformal truncation framework for infinite-volume dynamics

We study the momentum-space 4-point correlation function of identical scalar operators in conformal field theory. Working specifically with null momenta, we show that its imaginary part admits an expansion in conformal blocks. The blocks are polynomials in the cosine of the scattering angle, with degree ℓ corresponding to the spin of the intermediate operator. The coefficients of these polynomials are obtained in a closed-form expression for arbitrary spacetime dimension d > 2. If the scaling dimension of the intermediate operator is large, the conformal block reduces to a Gegenbauer polynomial C (d−2)/2 ℓ . If on the contrary the scaling dimension saturates the unitarity bound, the block is different Gegenbauer polynomial C (d−3)/2 ℓ

Section: The Scalar Two-point Functionmentioning

confidence: 99%

Momentum-space conformal blocks on the light cone

Gillioz

2018

“…We can bring the zero mode contributions back into view by starting with equal-time quantization and then taking the lightcone limit via an infinite boost. For this purpose, we 11 The questions raised by this discussion have broader implications that go beyond conformal truncation itself. If one can show that single-trace OPE data is sufficient to determine large N RG flows in a given class of theories, then all bulk contact interactions in the AdS duals of these theories, such as g 4 (or e.g.…”

Section: The Role Of Zero Modes In a Holographic Modelmentioning

confidence: 99%

“…Here we've used the fact that CFT spectral densities scale as ρ O (p) ∼ p 2∆−d 9. For more details in the particular case of a free theory, see[11], where the decoupling of zero modes due to IR divergences naturally led to a rearrangement of the naive conformal basis into new "Dirichlet" states with no overlap with zero modes.…”

mentioning

confidence: 99%

Lightcone effective Hamiltonians and RG flows

Fitzpatrick

Kaplan

Katz

et al. 2018

Self Cite

We present a prescription for an effective lightcone (LC) Hamiltonian that includes the effects of zero modes, focusing on the case of Conformal Field Theories (CFTs) deformed by relevant operators. We show how the prescription resolves a number of issues with LC quantization, including i) the apparent non-renormalization of the vacuum, ii) discrepancies in critical values of bare parameters in equal-time vs LC quantization, and iii) an inconsistency at large N in CFTs with simple AdS duals. We describe how LC quantization can drastically simplify Hamiltonian truncation methods applied to some large N CFTs, and discuss how the prescription identifies theories where these simplifications occur. We demonstrate and check our prescription in a number of examples. Appendix B. Details of the Bulk Model Old-Fashioned Perturbation Theory 52 Appendix C. SUSY Bulk Model 55 Appendix D. Lightcone Truncation and the Infinite Momentum Limit 56 References 61 The resulting theory can then be studied through non-perturbative Hamiltonian truncation techniques, which involve restricting the Hilbert space to a finite-dimensional subspace and numerically diagonalizing the truncated Hamiltonian exactly. Yurov and Zamolodchikov were the first to derive the low-lying spectrum of QFT using the truncated spectrum approach [1]. Recently, Hamiltonian truncation has been revived, in part thanks to several technical advancements that have improved the numerical predictivity of the method [2-5]. In the past few years Hamiltonian truncation has been applied with success to a variety of models, and to study several aspects of QFT, such as spontaneous symmetry breaking [3, 6, 7], scattering matrices [7], and quench dynamics [8]. 1 While many of the results and considerations in this paper should be generalizable to different UV bases, in this work we will focus on the particular implementation of conformal truncation [10, 11], which uses the eigenstates of the UV CFT Hamiltonian. These states can be organized into representations of the conformal group, each of which is associated with a primary operator O(x). Working in momentum space, we can write the states in the general form |O, P , µ ≡ d d x e −iP ·x O(x)|0 (µ 2 ≡ P 2 ). (1.2) These states are characterized by an eigenvalue C under the quadratic Casimir 2 of the 1 A more comprehensive list of references can be found in [9]. 2 This takes the familiar form C = ∆(∆ − d) + ( + d − 2) in terms of operator dimensions and spins.

“…This light front quantization [9] is also used in numerical solutions of strongly coupled QFTs via a version of HT; some recent work is [10][11][12][13][14][15]. The structure of the unperturbed Hilbert space is different from the equal time case, which leads to important differences in the numerical procedure.…”

Section: Jhep10(2017)213mentioning

confidence: 99%

High-precision calculations in strongly coupled quantum field theory with next-to-leading-order renormalized Hamiltonian Truncation

Elias-Miró

Vitale

Rychkov

2017

Hamiltonian Truncation (a.k.a. Truncated Spectrum Approach) is an efficient numerical technique to solve strongly coupled QFTs in d = 2 spacetime dimensions. Further theoretical developments are needed to increase its accuracy and the range of applicability. With this goal in mind, here we present a new variant of Hamiltonian Truncation which exhibits smaller dependence on the UV cutoff than other existing implementations, and yields more accurate spectra. The key idea for achieving this consists in integrating out exactly a certain class of high energy states, which corresponds to performing renormalization at the cubic order in the interaction strength. We test the new method on the strongly coupled two-dimensional quartic scalar theory. Our work will also be useful for the future goal of extending Hamiltonian Truncation to higher dimensions d 3. Keywords: Field Theories in Lower Dimensions, Nonperturbative EffectsArXiv ePrint: 1706.06121Open Access, c The Authors. Article funded by SCOAP 3 .https://doi.org/10.1007/JHEP10 (2017)213 JHEP10 (2017)213 Introduction. Many interesting strongly interacting Quantum Field Theories (QFTs) are not amenable to analytical treatment. Such theories are often studied via Lattice Monte Carlo (LMC) numerical simulations, starting from the discretized Euclidean action. However, LMC has some drawbacks, for example it cannot easily compute real-time observables, it is rather computationally expensive, and it cannot directly describe renormalization group (RG) flows starting from interacting fixed points. Therefore, it is worth exploring other numerical approaches to strongly interacting QFTs. One promising alternative is provided by the Hamiltonian methods, which look for the eigenstates of the quantum Hamiltonian. These methods use various finite-dimensional approximations to the full infinite-dimensional QFT Hilbert space. Notable examples are the methods using Matrix Product States [1, 2] and more general Tensor Networks [3] such as MERA [4] or PEPS [5]. In this paper we will be concerned with another representative of this group of methods -Hamiltonian Truncation (HT), also known as the Truncated Spectrum (or Space) Approach, which is a direct generalization of the variational Rayleigh-Ritz (RR) method from quantum mechanics. This method goes back to the seminal work of Yurov and Al. Zamolodchikov [6,7] and has since been applied in many contexts. See [8] for a recent extensive review and the bibliography.The idea of HT is simple. The QFT Hamiltonian operator H is split as H 0 + V where H 0 is an exactly solvable Hamiltonian whose eigenstates form the basis of the Hilbert space. One quantizes at surfaces of constant time and works in finite volume so that the spectrum is discrete. 1 The Hilbert space is then truncated to the low-lying eigenvectors of H 0 . The matrix of H in this truncated Hilbert space is diagonalized exactly on a computer, to find the low-energy spectrum of interacting eigenstates.As was understood early on [16], the numerical convergence of the HT...